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\begin{document}

\title{\vspace{-1\baselineskip}\textbf{Average Utility: A Preference Foundation\vspace{0\baselineskip}}}
\author{Amit Kothiyal, Vitalie Spinu, \& Peter P.\ Wakker \thanks{@}\\Econometric
  Institute, Erasmus University,\\P.O. Box 1738, Rotterdam, 3000 DR, the
  Netherlands\\~}
% @
\date{\vspace{0\baselineskip}July 9, 2011}
% \pagecolor[gray]{.03}
% \color[rgb]{0,1,0}

\maketitle

\begin{abstract} \vspace{0\baselineskip} This paper analyzes the evaluation of sequences of variable length through their average utility.  We use a new algebraic technique, exploiting the richness structure naturally provided by the variable lengths, and obtain necessary and sufficient conditions in full generality.  Thus we generalize preceding results in the literature.  For example, continuity in outcomes, a requirement in other approaches, now is an option rather than a requirement.  In our approach, continuity can be related to connected separable topologies, generalizing Euclidean topologies.  
  % Generalizations are also provided for @@
  Applications to von
  Neumann-Morgenstern expected utility and generalized mean functionals are also discussed.
\end{abstract}\vspace{0\baselineskip}


\section{Introduction}\label{Intr.Sect.1}

This paper examines sequences $(x_1, \ldots ,x_n)$ of varying length $n$, and provides a preference foundation for average utility evaluations ($\sum_{j=1}^{n}U(x_j)/n$).  This criterion describes the expected utility of lotteries with equal (thus including all rational) probabilities (von Neumann \& Morgenstern 1944), representative agent in utilitarian welfare evaluations with variable population size (Blackorby, Bossert, \& Donaldson 2005; Kolm 1969; Harsanyi 1953, 1955; Rawls 1971), the price index when different countries have different basic commodities (Balk 1995), the quality of life of different medical treatments tested on different samples (Drummond et al.\ 1997), the subjectively perceived nuisance of waiting times (Carmon, Shanthikumar, \& Carmon 1995), and so on.  Averages of transformed observations are relevant summary indexes in statistics (Norris 1986) and in many other contexts.

The main contribution of this paper is a new way of analyzing sequences of variable length.  Previous studies first used results for fixed length $n$, assuming continuity in outcomes to scale utility, and then extended their representation to variable lengths.\footnote{See 
  %%%%%%%%% 
  Blackorby, Davidson, \& Donaldson (1977 Lemma 2 and Assumption 5); Blackorby, Bossert, \& Donaldson (1999, p.\ 404 fifth paragraph and beginning of Section 5); Blackorby, Bossert, \& Donaldson (2005 Chs.\ 4-6); Gravel, Marchant, \& Sen (2008 following Theorem 1).}

We show that the variable lengths, available anyhow, in fact simplify the analysis.  They provide enough richness, and continuity is not needed.  With variable lengths we can define a concatenation operation that pastes sequences together, and then we use H\"{o}lder's (1901) simple but powerful lemma to cardinally scale utility.

Our preference conditions, provided in Section~\ref{sect.GenFound}, are necessary and sufficient in full generality, for arbitrary outcome sets, and generalize all previous result in the literature.  With continuity optional rather than required, our topological results in Section~\ref{sect.cont} are obtained for connected separable topologies instead of Euclidean topologies.  We also generalize mathematical axiomatizations of generalized (also called 
quasi-linear) means (Acz\'{e}l 1966; Hardy, Littlewood, \& Polya 1934; Kolmogorov 1930), again, by removing the requirement of continuity (Section~\ref{sect.gen.means}).

\section{General preference foundation for average utility}\label{sect.GenFound}

$X$ is a
non-empty {\it outcome space}.  It can be finite or infinite.  Greek letters $\ga, \xb, \ldots$ and indexed Roman letters $x_i$ denote outcomes.  {\it Prospects\/} are sequences of the form $x = (x_{1},...,x_{n}) \in X^n$, with $n \in \Na$.  $\#x = n$ is the length of the prospect.  The $x_j$s are sometimes referred to as {\it coordinates\/} of $x$.  $\SX = \cup{_{n \in \Na} X^n}$ is the set of all prospects.  That is, $n$ is variable and we consider finite sequences of any length.  The {\it concatenation\/}
$(x,y)$ denotes $(x_{1},\ldots,x_{n},y_{1},\ldots,y_{m})$; further, $1x=x$, $2x=(x,x)$, and $kx=(x,(k-1)x)$.

A preference relation $\p$ is given on $\SX$.  The notation $\succ, \rp, \srp, \sim$ is as usual.  A function $M$ {\it represents\/} $\p$ if $M : \SX \ra \R$ and $x \p y \Lra M(x) \geq M(y)$.  We then also say that $\p$ {\it maximizes\/} $M$, or that prospects are {\it evaluated\/} by $M$.  If a representing function exists then $\p$ is a {\it weak order\/}, i.e.\ it is {\it complete\/} ($x \p y$ or $y \p x$ for all $x,y$) and transitive.

We identify outcomes with the corresponding sequences of length 1.  The restriction of $\p$ to $X$ resulting this way is also denoted $\p$.  A function $U$ {\it represents\/} $\p$ {\it on\/} $X$ if $U : X \ra \R$ and $\ga \p \xb \Lra U(\ga) \geq U(\xb)$.

\begin{DEFINITION} {\it Average utility\/} ({\it AU\/}) holds if there exists
  $U: X \ra \R$ such that
  $$x \mo
  \frac{\sum_{i=1}^{\#x}U(x_{i})}{\#x}$$ represents $\p$.  $U$ is the {\it utility function\/}.  \ep
\end{DEFINITION}

AU implies that $U$ represents $\p$ on $X$.  It also implies {\it symmetry\/}:
$ (x_1,\ldots,x_n) \sim (x_{\pi(1)},\ldots,x_{\pi(n)})$ for all permutations $\pi$.  This condition is often called anonymity in welfare evaluations.  The following implication is characteristic of both additive and average utility.

\begin{DEFINITION} $\p$ satisfies {\it joint independence\/} if
  \begin{eqnarray} (c_1, x_2, \ldots, x_n) \p (c_1, y_2, \ldots, y_n) \Ra
    \nonumber \\ (d_1, x_2, \ldots, x_n) \p (d_1, y_2, \ldots, y_n) .~\square
  \end{eqnarray} 
\end{DEFINITION}
%%%%%%%%%%%%%%%% 
The condition implies that preferences between two $n$ tuples are independent of a common first coordinate.  Because of symmetry, preference then is independent of any common coordinate.  By repeated application, preference is independent of any number of common coordinates.  This condition is often called separability in the economic literature.  We do not use this term to avoid confusion with a topological condition of the same name defined later.

We often use the following generalization of joint independence: {\it Expansion independence\/} holds if
%%%%% 
\begin{equation}\label{exp.sep} x \p y \Lra (\ga, x) \p (\ga, y)
  {\rm ~whenever~} \#x = \#y.
\end{equation}
%%%%% 
In the presence of symmetry, the condition implies that inserting an extra common coordinate at any place does not affect preference.  By repeated application, inserting any number of common coordinates does not affect preference.  In general, this condition is somewhat stronger than joint independence (Lemma \ref{exp.sep.then.sep}) because it links preferences between prospects of different lengths.  In the presence of other conditions that we will use, mainly replication equivalence (which links preferences between prospects of different lengths), the two conditions become equivalent (Lemma \ref{sep.then.exp.sep}).\footnote{This
  %%%%%%%%% 
  equivalence is elementary in the sense that it does not use technical axioms such as Archimedeanity or continuity in its derivation.  All logical relations between intuitive conditions claimed in this paper will be elementary in this sense.}
%%%%%%%%%% 
In welfare evaluations, expansion independence implies that a new person better join a good population than an equally big bad population.  In a mathematical sense, expansion independence amounts to compatibility of preference and sequence concatenation.  Our application of H\"{o}lder's lemma will be based on this.

Because real numbers satisfy Archimedeanity, an Archimedean condition is necessary for any
real-valued model.  The following version will be used for average utility.  {\it Archimedeanity\/} holds if for all $x,y\in \SX$ with $\#x=\#y$ and $x\succ y$, and all $v,w \in \SX$ with $\#v=\#w$ $$(nx,v)\p (ny, w)$$ for some $n \in \Na$.  Lemma \ref{n.suff.large} will show in our setup that if the preference holds for some $n$ then it holds for all larger $n$, meaning that $n$ should be ``large enough.''

The conditions defined so far are satisfied both by additive and by average utility representations.  The following condition separates average from additive utility\footnote{Additive
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
  representations can be separated from average representations by a reinforced expansion independence where Eq.\ \ref{exp.sep} is also imposed for $x,y$ of different length (Pivato 2011; Wakker 1986).}:
%%%%%%%%%%%%%% 
\begin{DEFINITION} {\it Replication equivalence\/} holds if $x \sim mx$ for all $x,m$.  \ep
\end{DEFINITION}

The following theorem is our main result.  All the proofs are delegated to appendices.
\begin{THEOREM}\label{main.result} The following two statements are equivalent
  for $\p$ on $\SX = \cup_{n=1}^\iy X^n$.

  \begin{list} {(\roman{AA})}{\usecounter{AA}}
  \item There exists a utility function $U: X \ra \R$ such that $\p$ maximizes average utility: $(x_1,\ldots,x_n) \mo \frac{U(x_1) + \cdots + U(x_n)}{n}$.

  \item $\p$ satisfies the following five conditions:
    \begin{enumerate}

    \item Weak ordering;

    \item joint independence;

    \item symmetry;

    \item replication equivalence;

    \item Archimedeanity.
    \end{enumerate}
  \end{list} Further, the utility function $U$ in Statement (i) is unique up to level and unit.  \ep
\end{THEOREM}

The first four conditions in Statement (ii) (the Archimedean axiom not included) will often be called the {\it intuitive conditions\/}.  The rest of this section informally explains why our theorem can do without additional richness.


The only preference foundation of average utility that did not use continuity is, to the best of our knowledge, Fishburn (1972).  He considered subsets of a finite set rather than sequences as we do.  His preference conditions use a technique by Scott (1964), based on methods for solving linear inequalities.  As pointed out by Gravel, Marchant, \& Sen (2011 introduction), this leads to complex axioms.

Our main theorem combines the advantages of Fishburn's technique and the papers that used continuity.  We achieve complete generality as does Fishburn, leaving continuity optional.  At the same time, our intuitive axioms are the simplest and weakest ones used in the literature on continuous representations.

As suggested above, we will not first establish
fixed-finite dimensional results, but we immediately turn to general dimensions $n$.  To illustrate how we can scale cardinal utility in our general setup, assume outcomes $\xg \succ \xb \succ \ga$ and $n \elt \Na$ ($\succ$ denotes strict preference).  We can find $k$ such that the following preferences hold between three
$n$-tuples:
%%%%%%%%%% 
%% Can someone make braces?
\begin{equation}\label{preftoidentifyU}
  (\xg, \ldots, \xg, \ga, \ga, \ldots, \ga) //rp (\xb, \ldots, \xb) //rp (\xg, \ldots, \xg, \xg, \ga, \ldots, \ga).
\end{equation}
%%%%%%% 
Here, in the left and right
$n$-tuples the first $k$ elements, the $k+1$th element, and the remaining $n - k - 1$ elements are displayed.  These preferences revealg
%%%%%%%%% 
\begin{equation}\label{revealedU} \frac{k}{n} < \frac{U(\xb) - U(\ga))}{U(\xg) - U(\ga))} < \frac{k+1}{n} .
\end{equation}
%%%% 
In plain words, we simply count how many advantages $\xg \succ \xb$ it takes to offset a number of disadvantages $\ga \srp \xb$.
Because $n$ can be taken arbitrarily large, we can identify cardinal utility as accurately as we want.  This is why we can do without the richness assumption of a continuum domain.  The intuitive preference conditions can be interpreted as consistency conditions for utility measurements.  The latter should not be affected by permutations (symmetry), the addition of common coordinates (expansion independence), or replications (replication invariance).

\section{Preference foundation for continuous average utility}\label{sect.cont}

This section assumes that $X$ is a topological space.  It is {\it connected\/} if the only sets that are both open and closed are $\emptyset$ and $X$.  It is {\it separable\/} if there exists a countable dense subset.  {\it Dense\/} means that every nonempty open set contains an element of the subset.  To obtain continuous representations, we use a continuity condition introduced by Gravel, Marchant, \& Sen (2008), because it is the weakest condition used in the literature, leading to the strongest theorems.  It is a remarkable weakening of the simple continuity (continuity with respect to 
finite-dimensional product topologies; see Section \ref{alt.pref.found}) assumed in all other papers in the literature.
% Ook Hardy, Hardy, Littlewood \& P\'{o}lya (1934 Theorem 215)

\begin{DEFINITION} $\p$ on $\SX$ is {\it continuous\/} if the sets $\{\ga \elt X: \ga \succ x\}$ and $\{\ga \elt X: \ga \srp x\}$ are open for every $x\in \SX$.  \ep
\end{DEFINITION}

\begin{THEOREM}\label{main.result.cont} The following two statements are equivalent for $\p$ on $\SX = \cup_{n=1}^\iy X^n$.

  \begin{list} {(\roman{AA})}{\usecounter{AA}}
  \item There exists a utility function $U: X \ra \R$ such that $\p$ maximizes average utility:\\ $(x_1,\ldots,x_n) \mo \frac{U(x_1) + \cdots + U(x_n)}{n}$\\
    with $U(X)$ an interval.

  \item $\p$ satisfies the following five conditions:
    \begin{enumerate}

    \item Weak ordering;

    \item joint independence;

    \item symmetry;

    \item replication equivalence;

    \item continuity with respect to a connected and separable topology on $X$.
    \end{enumerate}
  \end{list} Further, the utility function $U$ in Statement (i) is continuous (with respect to the topology\footnote{We
    %%%%%%%%%%%%% 
    will show in Lemmas \ref{can.do.order.top.1} and \ref{can.do.order.top.2} that this topology has to be a refinement of the order topology generated by $\p$ on $X$.  These lemmas also show that it can be any such refinement.  Lemmas \ref{cont=CE} and \ref{U.noncont.gap} further illustrate that the choice of the refined topology is immaterial.  This also holds for the other results in this section.}
  %%%%%%%%%%% 
  in point (5) in Statement (ii)), and it is unique up to level and unit.  \ep
\end{THEOREM}

\noi Theorem \ref{main.result.cont} differs from Theorem \ref{main.result} in the following ways.  In Statement (i) we added that $U(X)$ is an interval\footnote{Equivalently, 
  %%%%%%%%%%%%% 
  there exists a connected topology with respect to which $U$ is continuous.  This topology can always be taken separable).}
%%%%%%%%%%%%% 
% rrr on preceding
In Statement (ii) we replaced Archimedeanity with the stronger requirement of continuity of $\p$.

The topological requirements in the theorem are satisfied, for instance, if $X$ is a convex subset of a Euclidean space.  Then, given the other conditions, $\p$ is continuous if and only if $U$ is, and the topology can be taken to be the usual Euclidean one.  This is the most common case in economics.

Continuity of $\p$ when restricted to the outcome set $X$ is a relatively weak condition.  It is, for instance, satisfied whenever $X = \R$ and $\p$ is monotonic.
Continuity of $\p$ on the domain of all prospects, $\SX$, is stronger because it adds the existence of certainty equivalents.  We call $\ga\in X$ a {\it certainty equivalent\/} ({\it CE}) of $x\in \SX$ if $\ga \sim x$.  In general, a prospect $x$ can have no CE, or many CEs, which then are all equivalent.  The {\it CE condition\/} holds if every prospect has a CE.

\begin{LEMMA}\label{cont=CE} Assume the intuitive conditions.  Continuity of $\p$ on the outcome set $X$ w.r.t.\ a connected topology and the CE condition are equivalent to continuity of $\p$ on the set of prospects $\SX$ with respect to a connected topology on $X$.
  \ep
\end{LEMMA}


\section{Expected utility for decision under risk}\label{sect.DUR}

This section applies the preceding theorems to decision under risk, providing alternative derivations of expected utility.  It shows, in fact, how H\"{o}lder's lemma can be used to prove the von
Neumann-Morgenstern expected utility theorem.
We now interpret
$n$-tuples as $1/n$ probability distributions over outcomes.  Average utility then is expected utility.  Our domain in fact contains every simple probability distribution using only rational probabilities (Blackorby, Davidson, \& Donaldson 1977 p.\ 354; Grabisch et al.\ 2011b Section 2.3).  For example, the probability distribution $(2/5:15,3/5:0)$, where notation is as usual, corresponds with the 
equally-probable five-tuple $(15,15,0,0,0)$.  Thus our theorems axiomatize expected utility on simple
rational-probability prospects, with general utility in Theorem \ref{main.result} and with continuous utility in Theorem \ref{main.result.cont}.

Joint independence is implied by a von Neumann \& Morgenstern independence condition often used in decision under risk.  When all weighted 
$n$-tuples (simple probability distributions) $(p_1:x_1, \ldots, p_n:x_n)$, denoted $C, P, Q$ and so on, are incorporated in the domain. 
{\it NM-independence} requires that
%%%%%%%% 
\begin{equation}\label{vNMindep} P \p Q \Lra \gl C + (1- \gl)P \p \gl C + (1-\gl)Q \text{~for~all~} 0 < \gl < 1,
\end{equation}
%%%%%%% 
with the probabilistic mixture operation defined the usual way. 
{\it Restricted NM-independence\/} imposes the condition only on 
rational-probability distributions, which in particular concerns only rational $\gl$.

\begin{LEMMA}\label{ind=>joint.ind} Under weak ordering, 
  NM-independence on the set of weighted 
  $n$-tuples implies joint independence on the subset of unweighted 
  $n$-tuples.  \ep
\end{LEMMA}


In decision under risk, replication equivalence and symmetry are satisfied by definition because they then are different ways of writing the same probability distribution.  Theorem~\ref{main.result} and Lemma~\ref{ind=>joint.ind} give the following result:


\begin{COROLLARY}\label{th:NM} Expected Utility holds on the set of simple
  rational-probability prospects if and only if $\p$ is an Archimedean weak order satisfying 
  NM-independence.
  %%% 
\end{COROLLARY}

Additional axiom such as monotonicity is needed to assure the expected utility representation on the set of all probability distributions.

Similarly, it is easy to obtain subjective expected utility for uncertainty if we relate 
$n$-tuples to Savage's (1954) uniform partitions.  For instance, joint independence is equivalent to Savage's (1954)
sure-thing principle (preference being independent from common outcomes).


\section{Generalized mean functionals}\label{sect.gen.means}

Foundations of average utility representations can be obtained if preference theory is linked
to the mathematical theory of generalized means\footnote{
  %%%%%% 
  This link, explained in this section, has been used by several authors (Bullen 2003; Chew 1983; Grabisch et al.\
  2009, 2011a, 2011b p.\ 42; Muliere \& Parmigiani 1993; Ozaki 2009; Wakker 1988). }
%%% 
. The latter is a subfield of functional equations (Acz\'{e}l 1966).  Kolmogorov (1930) and Naguma (1930) were the first to, independently, provide axiomatic foundations of generalized means.  Modifications and generalizations include Grabisch et al.\ (2009) and Marichal (2000).


$M$ is a {\it generalized mean\/} (Bonferroni 1924) if there exists a strictly increasing function $U$ %%%%%%%% .
%%%%%%%%% 
such that
%%%%%%%%%% 
\begin{equation}\label{ordinalCEAU} M(x_1,\ldots, x_n) = U^{-1} (\sum U(x_j)/n)
\end{equation} 
with $U^{-1}$ always well defined.


In the theory of generalized means a function $M : \SX \ra \R$ is taken as primitive\footnote{Taking a function, rather than an ordering, as primitive, is common in many fields in economics, including production theory (Nicholson 2005), and the study of price indexes (Balk 1995) and risk indexes (Artzner et al.\ 1999).} and its form is characterized through necessary and sufficient conditions. {\it Monotonicity\/} - $M$ is strictly increasing in each coordinate - is one such condition. A common  assumption in this field is that  $X \sbs \R$ is an interval, which we also require. We summarize the assumptions of this section:

\begin{ASSUMPTION} [Structural assumption] \label{str.ass.M.}  $X \sbs \R$ is an interval, $\SX = \cup{_{n \in \Na} X^n})$, and $M:\SX \ra \R$ is monotonic.  $\p$ denotes the preference relation on $\SX$ represented by $M$.  \ep
\end{ASSUMPTION}

We first give direct reformulations of our AU preference conditions for the function $M$.  $M$ satisfies:
\begin{itemize}
\item {\it reflexivity\/} if $M(\ga) = \ga$ for all outcomes $\ga$;

\item {\it symmetry\/} if $M$ is invariant under permutations of coordinates;

\item {\it joint independence\/} if $M(c_1,x_2,\ldots,x_n) \geq
  M(c_1,y_2,\ldots,y_n) \Ra$ \\ $M(d_1,x_2,\ldots,x_n) \geq
  M(d_1,y_2,\ldots,y_n)$;

\item {\it replication equivalence\/} if $M(x) = M(nx)$ for all $n,x$.
\end{itemize}

For our purposes it is convenient not to require continuity of $U$ from the outset, although it follows from the other conditions (Corollary \ref{gen.M=>cont}) \footnote{$U$, called {\it utility function\/} in this paper, is often called the generator of $M$ in the literature (Grabisch et al.\ 2011a).  Other terms for generalized means are extended
  quasi-arithmetic means (Grabisch et al.\ 2011a) and symmetric
  quasi-linear means (Acz\'{e}l 1966).}. 


\begin{THEOREM}\label{generd.means} [Characterizing generalized means].  Assume Structural Assumption \ref{str.ass.M.}.  Then the following two statements are equivalent:
  \begin{list} {(\roman{AA})}{\usecounter{AA}}
    % 
  \item $M$ is a generalized mean: $M = U^{-1} (\sum U(x_j))$ for a strictly increasing $U$.
    % 
  \item $M$ satisfies reflexivity, symmetry, joint independence, and replication equivalence.
  \end{list} $U$ in (i) is continuous and it is unique up to level and unit.
  \ep
\end{THEOREM}
%%%%%%%%% 

The above theorem generalizes results by Kolmogorov (1930) and Nagumo (1930).  These authors used an {\it associativity\/} condition:
%%%%%%%%%%%%% 
\begin{equation} M(x_1,\ldots, x_k) = \ga \Ra M(x_1,\ldots, x_k, x_{k+1}, \ldots, x_n) = M(k\ga, x_{k+1}, \ldots, x_n).
\end{equation}
% 
Here $k\ga$ refers to
$k$-fold concatenation, and not to multiplication.  Associativity can be equated with the substitution principle of decision under risk, which is a weak version of the usual independence condition.  It allows replacement of a conditional lottery by an equivalent other conditional lottery (such as its CE) without affecting preference.  Blackorby, Bossert, \& Donaldson's (2005 p.\ 125) population substitution principle, when imposed on their representative agent, is equivalent.  Kolmogorov (1930) and Nagumo (1930) also used {\it idempotence\/}, a reinforcement of reflexivity: $M(n\ga) = \ga$.\footnote{Some
  %%%%%%% 
  papers interchange the terms idempotence and reflexivity.}
%%%%%%% 
It can be seen, under symmetry, that associativity is equivalent to joint independence and replication equivalence, and that it implies idempotence.  We only prove the implications needed for our analysis.  Reversed implications can be derived from Lemma \ref{strong.ass}, but we will not elaborate on this.

\begin{LEMMA}\label{ass=>} Assume Structural Assumption \ref{str.ass.M.} and symmetry.  Then associativity implies idempotence, reflexivity, replication equivalence, and joint independence.  \ep
\end{LEMMA}

Thus the theorems of Nagumo and Kolmogorov are obtained as corollaries of Theorem \ref{generd.means}.  Remarkably, Nagumo and Kolmogorov assumed continuity of $M$, but this assumption can be dropped.  It is implied by the other assumptions, as Theorem \ref{generd.means} showed.  The following corollary displays this point, showing that our continuity (taken from Gravel, Marchant, \& Sen 2008) entails no restriction for the study of generalized means.

\begin{COROLLARY}\label{gen.M=>cont} Assume Structural Assumption \ref{str.ass.M.} and reflexivity, symmetry, joint independence, and replication equivalence.  Then the generated $\p$ on $\SX$ is continuous.  In particular, $\p$ and $U$ are continuous if $M$ is a generalized mean.  \ep
\end{COROLLARY}

Several authors studied generalized means for weighted prospects (de Finetti 1931; Hardy, Littlewood \& P\'{o}lya (1934 Theorem 215).  Then the {\it generalized mean\/} is defined as
$U^{-1}\sum_{j=1}^n p_jU(x_j)$.  An obvious interpretation is that the prospects are probability distributions and the generalized mean is the CE under expected utility, as in Section \ref{sect.DUR}.  Axiomatizations similar to those of Naguma (1930) and Kolmogorov (1930) were given, using a modified associativity condition:
%%%%%%%%%%%%%%% 
\begin{equation}\label{vNMindepsubs} M(P) = M(Q) \Lra M(\gl C + (1- \gl)P) = M(\gl C + (1-\gl)Q) \text{~for~all~} 0 < \gl < 1.
\end{equation}
%%%%%%%%% 
The condition implies associativity because we can apply it to the case where $Q$ is the CE of $P$, and then it implies that a conditional subpart of a prospect can be replaced by its CE.  The condition is a weakened version of the independence condition (Eq.\ \ref{vNMindep}) because it only considers indifferences/equalities.  Our Theorem \ref{generd.means} can be used here similarly as in Section \ref{sect.DUR}, with our domain containing all prospects with rational weights.  In the context of weighted 
$n$-tuples, Hardy, Littlewood, \& P\'{o}lya (1934 Theorem 215) noticed that continuity of the function $M$ need not be imposed because it is implied by the other conditions, similarly as in our results.


\section{General discussion and related literature}\label{lit}

\subsection{The problematic empirical status of continuity}

Besides intuitive conditions typical of the model considered, preference foundations usually adopt richness assumptions implying that the domain is a continuum.  Such richness greatly facilitates the scaling of utility and simplifies the intuitive preference conditions that are needed.  Thus with one exception discussed later, all analyses of average utility that we are aware of assumed a continuum of outcomes with continuous utility.  

Although continuity brings many advantages and is often appealing (Blackorby, Bossert, \& Donaldson 2005 p.\ 71; Chew 1983 p.\ 1070), there are empirical problems.  Unlike what has sometimes been thought, continuity is not merely a technical condition, but it adds empirical content to the other conditions.  What makes this point problematic is that it is usually not clear what that empirical content is.  Then we do not know well what our models mean.  Fuhrken \& Richter (1991) investigated this problem for general additive representations, where they could identify the added empirical content of utility.  This added content consisted of an infinite list of axioms that cannot be formulated in
first-order logic (Scott \& Suppes 1958).  Shapiro (1979) investigated the same question for Savage's (1954) subjective expected utility.  He obtained very complex conditions, for which it is hard to understand their empirical meaning.  Pfanzagl (1968 Section 6.6, pp.\ 107-108, Remark on p.\ 111, and Section 9.5) provided a general formal analysis of this problem, and Krantz et al.  (1971 Section 9.1) discussed it too.

As regards empirical meaning, the Archimedean axiom fares better than continuity.  Archimedeanity is usally purely technical in that it does not add empirical content to the other axioms (Luce et al.\ 1990 Theorem 21.21; Fuhrken \& Richter 1991 p.\ 94).  It only serves to reduce 
non-standard real representation to standard real numbers.  Thus the empirical meaning of Archimedeanity is clear: it is void.  And thus the Archimedean axiom is not problematic.  For further discussions of the Archimedean axiom, see Narens (1985), Luce et al.\ (1990 Ch.\ 21), and Suppes 1974 p.\ 164).

\subsection{Additive and average utility}

The two most common ways to subjectively evaluate sequences are by sums or by averages of their utilities.  These two ways generate the same preferences between 
$n$-tuples if $n$ is fixed.  They part ways if $n$ is variable and if then sequences of different length are compared with each other.  The latter comparisons are the topic of this paper.  

Additive representations have been extensively studied, and numerous preference foundations have been provided, often based on bisymmetry conditions (Fishburn 1986, 1992; Fuhrken \& Richter 1991; Krantz et al.\ 1971; Wakker 1986, 1989).  Pivato (2011) provided advanced results for sequences of variable length, with extensions to infinitesimal representations without Archimedeanity (see also Fuhrken \& Richter 1991 p.\ 84).
Barber\'{a} \& Pattanaik (1984) and Kannai \& Peleg (1984) give axioms to separate additive and average evaluations.  We will not review the large literature on additive representations in detail, and focus this paper on average representations for variable lengths.
% Dit houdt resultaten met bisymmetry van Acz\'{e}l e.d. buiten de
% deur(Peter).

\subsection{Alternative preference foundations of average utility}\label{alt.pref.found}

Despite the widespread use of average utility, there have not yet been very many preference foundations in the literature.  Section~\ref{sect.gen.means} demonstrated how foundations can be obtained if preference theory is linked to the mathematical theory of generalized means.  We discussed Fishburn (1972)'s foundation in Section \ref{sect.GenFound}.  This section discusses some other contributions. All references discussed assume weak ordering and we also assume it throughout this section.  

Many papers assume, instead of Gravel, Marchant, \& Sen's (2008) continuity which we assumed, the following stronger condition.  {\it Simple continuity\/} holds if, for each $n$, the restriction of $\p$ to $X^n$ is continuous with respect to the product topology.  In the presence of the requirement $\ga \sim k\ga$ as implied by replication equivalence, this condition implies our continuity.

Instead of our replication equivalence\footnote{Their
  %%% 
  term replication equivalence is similar to what we call idempotence.},
%%%%%%%%%%%%%%%%% 
Gravel, Marchant, \& Sen (2008) used a stronger {\it averaging\/} preference condition: $x \p y \Ra x \p (x,y) \p y$.

\begin{LEMMA}\label{av.=>repl.eq} Under weak ordering, averaging implies replication equivalence.  \ep
\end{LEMMA}

Their {\it same number enlargement} and {\it same number existence consistency} are equivalent to expansion independence, which in turn implies our joint independence (Lemma \ref{exp.sep.then.sep}).  Their {\it existence of critical levels} is, in the presence of some other conditions such as averaging, equivalent to our CE condition, which implies our continuity (Lemma \ref{cont=CE}).
%% 

Their main result, Theorem 2, shows that an AU representation exists for $X = \R^k$ if weak ordering, symmetry, averaging, expansion independence, continuity, and minimal increasingness (a monotonicity condition on $\R^k$) hold.  Because $\R^k$ is a connected separable topological space, their conditions imply the conditions of our Theorem \ref{main.result.cont} and their result follows from our theorem.

Blackorby, Davidson, \& Donaldson (1977) assumed $X = \R_{+}^n$.  Their Assumption 3 is simple continuity, implying our continuity.  Their Assumption 4 is joint independence.\footnote{Their
  %%%% 
  Lemma 1, with $U^s$ dependent on the length $s$ of the prospects, suggests that their Assumption 4 is not meant to capture expansion independence.}
%%%%%%%%%% 
Symmetry is their Definition 1.  Replication equivalence is an implication of their implicit assumption (in their Lemma 2) that a prospect is sufficiently described by the subjective probability distribution that it generates over outcomes, a condition sometimes called probabilistic sophistication (Machina \& Schmeidler 1992).  Thus their main result, Theorem 1, follows from our Theorem \ref{main.result.cont}.
% Their Section IV briefly discusses a restriction of the AU representation
% to nonconstant prospects, i.e., with prospects $k\ga$ excluded.  Preference 
% foundations for this model are in Diecidue, Schmidt, \& Wakker (2004),
% Fishburn (1980), and Luce et al.\ (2008).

Blackorby, Bossert, \& Donaldson (2005 Theorem 6.15 and the text following the theorem) assume $X = \R$.  Outcomes are interpreted as individual utilities, assumed available as observed inputs.  The authors assume simple continuity, Pareto weak preference and minimal increasingness (two monotonicity conditions), replication equivalence, symmetry (see their p.\ 198 line 2, ``anonimity''), and
same-people independence which is our joint independence (see their p.\ 198) to obtain AU.  Again, all conditions in our Theorem \ref{main.result.cont} are satisfied and their result follows from our result.

Gravel, Marchant, \& Sen (2011) considered AU maximization over sets instead of sequences.  We focus on sequences, which are an essentially different domain.  Hence there is no logical relation between their result and ours.

\subsection{Different domains and more general functionals}

There exist many preference foundations of models that generalize average utility by bringing in other subjective factors, such as nonequal subjective weightings of coordinates that may be additive\footnote{Such
  %%%%%%%%%%% 
  models include subjective expected utility (Ahn 2008; Bolker 1967; Gul 1992; Jeffrey 1965; Ramsey 1931; Savage 1954; Wakker 1989), discounted utility (Bleichrodt, Rohde, \& Wakker 2008; Koopmans 1960, 1972; Kopylov 2010), and utilitarian welfare (Blackorby, Bossert, \& Donaldson 2005; Harsanyi 1955; Kolm 2002).}
%%%%%%%%%%%%% 
or nonadditive\footnote{Such
  %%%%%%%%%% 
  models include
  rank-dependent utility (Gilboa 1987; Grabisch et al.\ 2009, 2011a Section 4, 2011b; Quiggin 1982; Schmeidler 1986), prospect theory (Tversky \& Kahneman 1992), betweenness models (Chew 1983; Dekel 1986; Fishburn 1986; Ozaki 2009), the Sugeno integral (Chateauneuf, Grabisch, \& Rico 2008; Sugeno 1974), and other models (Jaffray 1989; Olszewski 2007).}.
%%%%%%%%%%%%%%%% 
In many of these, our finite 
$n$-tuples of outcomes are replaced with continuous streams of outcomes.  In all these cases, our equally weighted average utility is present as a special case in a substructure.  There is, however, no easy way to obtain preference foundations of unweighted average utility from those more general models.  They derive weaker conclusions from weaker axioms and essentially use the extra structure in their derivations.  Hence these results are logically independent of the results of this paper.

@Following para to be tuned in with the risk section.
Several papers on generalized means considered an extended domain of {\it weighted prospects\/}, i.e.\ 
$n$-tuples $(p_1:x_1, \ldots, p_n:x_n)$ with the $p_j$s nonnegative and summing to 1, where $n$ may vary (Chew 1983; de Finetti 1931; Hardy, Littlewood, \& Polya 1934 Section 6.20; Muliere \& Parmigiani 1993).  The $p_j$s may be irrational.  Weighted prospects can for instance designate simple probability distributions in decision under risk.  We discussed the extended domain in Section \ref{sect.DUR} and at the end of Section \ref{sect.gen.means}.


\section{Conclusion} 
We have studied average utility representations on sequences of varying length.  Our main contribution is to demonstrate how the richness generated by the varying length can be exploited to simplify the analysis.  This richness allows for a concatenation operation on sequences to which H\"{o}lder's (1901) Lemma can be applied.  We thus obtain necessary and sufficient conditions for average utility in full generality, and we generalize all results in the literature.  In particular we show that continuity in outcomes, assumed in preference representations and in the mathematical theory of generalized means, can be generalized and often is redundant.  An obvious topic for future research concerns the use of our technique, exploiting the richness of varying lengths, to analyze other functionals than AU, including many other welfare criteria for variable population sizes (Blackorby, Bossert, \& Donaldson 2005; Grabisch et al.\ 2009, 2011a, 2011b).

\vspace{2\baselineskip}

\noi {\Large \bf Appendix A: Preparatory results and proofs without continuity}\vspace{\baselineskip}

\begin{LEMMA}\label{exp.sep.then.sep} Under weak ordering, expansion independence implies joint independence.  \ep
\end{LEMMA}

\noi {\sc Proof.}  Assume $(x_2,\ldots, x_n) R (y_2,\ldots, y_n)$, with either $R =$ $\succ$, or $R =$ $\sim$, or $R =$ $\srp$.  By expansion independence, both $(c_1, x_2,\ldots, x_n) R (c_1, y_2,\ldots, y_n)$ and $(d_1, x_2,\ldots, x_n) R (d_1, y_2,\ldots, y_n)$.  Hence the latter two relationships are always the same, implying joint independence.  \ep\vspace{\baselineskip}

\begin{LEMMA}\label{sep.then.exp.sep} The intuitive conditions imply expansion independence.  \ep
\end{LEMMA}

\noi {\sc Proof.}  Assume an outcome $\ga$ and $\#x = \#y = n$.  Take some $c$ with $\#c = n^2$ and assume the relationship $(x,c) R (y,c)$ where either $R =$ $\succ$, or $R =$ $\sim$, or $R =$ $\srp$.  We will show below that both $x R y$ and $(x,\ga) R (y,\ga)$.  That is, the latter two relationships are the same whatever they are.  This implies expansion independence.

By joint independence (applied $n^2$ times) we have $(x,d) R (y,d)$ for all $d$ of length $n^2$.  In words, replacing an 
$n$-tuple $x$ by an 
$n$-tuple $y$ in any $n(n+1)$ tuple generates an $R$ relation.

We, hence, have $((j+1)x,(n-j)y) R (jx,(n+1-j)y)$ for all $j$, which by transitivity implies $(n+1)x R (n+1)y$.  By replication equivalence, it gives $x R y$.

We also have $((j+1)x,(n-j-1)y,n\ga) R (jx,(n-j)y,n\ga)$ for all $j$, which by transitivity implies $(nx, n\ga) R (ny, n\ga)$.  By replication equivalence, $(x.\ga) R (y,\ga)$.  \ep\vspace{\baselineskip}

{\it Monotonicity\/} holds if weakly improving an outcome in a prospect gives a weakly preferred prospect, and strictly improving an outcome in a prospect gives a strictly preferred prospect.  Repeated application of the condition implies the same if several outcomes are replaced.

\begin{LEMMA}\label{monotonicity} The intuitive axioms imply monotonicity.
  \ep
\end{LEMMA}

\noi {\sc Proof}.  Let $R =$ $\p$ or $R =$ $\succ$.  By expansion independence applied $n-1$ times (for $x_2,\ldots, x_n$), $\xb R \ga$ implies $(\xb, x_2,\ldots, x_n) R (\ga, x_2,\ldots, x_n)$.  \ep\vspace{\baselineskip}

{\it Strong associativity\/} holds if
\begin{equation} (x_1,\ldots, x_k) \p (y_1,\ldots, y_k) \Lra (x_1,\ldots, x_k, c_{k+1}, \ldots, c_n) \p (y_1,\ldots, y_k, c_{k+1}, \ldots, c_n) .
\end{equation}
% 
In words, improving a subprospect improves the whole prospect.  The condition reinforces monotonicity by considering subprospects of length exceeding 1.

\begin{LEMMA}\label{strong.ass} Expansion independence implies strong associativity.  \ep
\end{LEMMA}

\noi {\sc Proof}.  Apply expansion independence $n-k$ times (for $c_{k+1}, \ldots, c_n$).  \ep\vspace{\baselineskip}

The following result concerns Archimedeanity.

\begin{LEMMA}\label{n.suff.large} Assume the intuitive axioms, $x \succ y$, and $(nx, v) \p (ny,w)$.  Then $(mx, v) \p (my,w)$ for all $m \geq n$.  \ep
\end{LEMMA}

\noi {\sc Proof}.  By applying strong associativity twice, we get $((m+1)x, v) \p (y,mx, v) \p (y, my, w) = ((m+1)y, w)$.  \ep\vspace{\baselineskip}


\noi {\sc Proof of Theorem \ref{main.result}.}  Necessity of the five conditions in Statement (ii) follows from substitution.  We, hence, assume the five conditions of Statement (ii) and derive the AU representation (and after that establish the uniqueness result).  We will use Theorem 3.2.1.1 of Krantz et al.\ (1971)\footnote{This
  %%%%%%%%%%% 
  is H\"{o}lder's (1901) lemma without the requirement that every element have an inverse.},
%%%%%%%%%%% 
and will verify its conditions.  Our concatenation operation of prospects corresponds with the concatenation operation denoted $\circ$ by Krantz et al.  We define a binary relation $\p^*$ on prospects that will turn out to be the additive counterpart of the ``averaging'' binary relation $\p$.  To this effect, we take an arbitrary outcome $\theta \in X$.  We will let $\theta$ play the role of a neutral element with respect to the concatenation (``addition'') and $\p^*$, later setting $U(\theta) = 0$.

\begin{DEFINITION}\label{deff.addpref} $x \p^* y$ if there exists $m \geq max\{\#x, \#y\}$ such that $(x, (m-\#x)\theta) \p (y, (m-\#y)\theta)$.
\end{DEFINITION}

The preference $\p$ in Definition \ref{deff.addpref} refers only to prospects of the same length, in which case additive and average utility give the same result.  If there exists an $m \geq max\{\#x, \#y\}$ such that the preference in Definition \ref{deff.addpref} holds then, by expansion independence (Lemma \ref{sep.then.exp.sep}), the preference holds for all $m \geq max\{\#x, \#y\}$.  Roughly, $\p^*$ is derived from $\p$ by starting from
equal-length comparisons, and then adding or deleting $\theta$s as is desirable.  The symmetric part $\sim^*$ and the asymmetric part $\succ^*$ are defined as usual.  We verify that $\p^*$ satisfies all four conditions of Theorem 3.2.1.1 of Krantz et al.\ (1971).  For conditions 2-4 we first give their definition, and then their derivation.

\begin{enumerate}
\item (Weak ordering).  Completeness of $\p^*$ follows immediately from completeness of $\p$.  For transitivity, assume that $x \p^* y$ and $y \p^* z$.  Then $(x,m-\#x)\theta \p (y,m-\#y)\theta \p (z,m-\#z)\theta$ for all $m \geq max\{\#x, \#y, \#z\}$, implying, by transitivity of $\p$, $(x,m-\#x)\theta \p (z,m-\#z)\theta$ for all such $m$.  This implies $x \p^* z$, and transitivity follows.

\item (Weak associativity): $(x,(y,z)) \sim^* ((x,y),z)$.  This follows from idempotence because we even have equality here.  (Our concatenation satisfies what is sometimes called associativity for operations.)

\item (Krantz et al.\  monotonicity): $x \p^* y \Lra (c,x) \p^* (c,y) \Lra (x,c) \p^* (y,c)$.  This term of Krantz et al.\ deviates from our term monotonicity.  It is in fact expansion independence extended to prospects $x,y$ of different length.  $(x, (m-\#x)\theta) \p (y, (m-\#y)\theta)$ is, by expansion independence, equivalent to $(z,x, (m-\#x)\theta) \p (z,y, (m-\#y)\theta)$, which gives the first logical equivalence in monotonicity.  The second logical equivalence follows from symmetry.

\item (Archimedeanity): If $x\succ^* y$, then for all $v,w \in \SX$ there exists an $n$ such that $(nx,v) \p^* (ny,w)$.  Assume that $x\p^* y$, that is, $(x, (m-\#x)\theta) \p (y, (m-\#y)\theta)$.  By Archimedeanity of $\p$, $(n(x, (m-\#x)\theta),v) \p (n(y, (m-\#y)\theta),w)$ for some $n$.  Dropping all the $\theta$s we get $(nx,v) \p^* (ny,w)$, as required.
\end{enumerate}

All conditions in Krantz et al.\ (1971, Theorem 3.2.1.1) are satisfied.  Hence there exists a real valued function $U(\cdot)$ unique up to a positive scale factor, that is additive with respect to the concatenation operation ($U(x,y)=U(x)+U(y)$).\footnote{This
  %%%%%%%%% 
  result for $\p^*$ generalizes Blackorby, Bossert, \& Donaldson (2005 Theorems 4.21 and 4.22) by dropping continuity of the representation, and weakening the continuity preference condition to Archimedeanity.}
%%%%%%%%%%%%% 
We get $U(\theta) = 0$.  The idea underlying their proof is to, first, define $U$ through Eqs.\ \ref{preftoidentifyU} and \ref{revealedU} and limit taking for $n\ra \iy$.  In these equations we only compare sequences of the same length, implying that the preferences $\p$ agree with $\p^*$.  We obtain $U$ correctly irrespective of whether the representation is additive, as with $\p^*$, or multiplicative, as with $\p$.  The preference conditions imply that the revelations of utility do not generate inconsistencies.  The conditions of Theorem 3.2.1.1 of Krantz et al., similar to H\"{o}lder (1901), imply that sums of $U$ values represent $\p^*$.  Mostly their monotonicity condition for $\p^*$ implies that its representation is additive and not average.

For $\p$ we do not have monotonicity in the Krantz et al.\ sense, but replication equivalence.  We finally show that this implies that averages of $U$ represent $\p$.  Because of replication equivalence, we have $x \p y \Lra (\#y)x \p (\#x)y$.  Because the latter preference concerns two prospects of the same length, it is equivalent to $(\#y)x \p^* (\#x)y$, or $(\#y)\sum_{i=1}^{\#x}U(x_i) \geq (\#x)\sum_{j=1}^{\#y}U(y_j)$.  This is equivalent to $\frac{\sum_{i=1}^{\#x}U(x_{i})}{\#x}\geq \frac{\sum_{i=1}^{\#y}U(y_{i})}{\#y}$.  The AU representation of $\p$ has been derived.

We finally consider the uniqueness result.  Substitution immediately shows that we are free to choose level and unit of $U$.  To see that there is no other liberty, consider $U(\theta)$ for some $\theta$ and define $\p^*$ as above with respect to $\theta$.  The above proof has demonstrated, using Theorem 3.2.1.1 of Krantz et al.\ (1971), that only the unit of $U$ then can be changed.  \ep\vspace{\baselineskip}



\noi {\Large \bf Appendix B: Preparatory results and proofs with continuity (Section \ref{sect.cont})}\vspace{\baselineskip}

\begin{DEFINITION}\label{order.top} For the preference relation $\p$ on $X$, the {\it order topology\/} is the smallest topology that makes $\p$ {\it continuous\/}, i.e.\ that contains all sets $\{\ga \in X: \ga \succ \xb\}$ and $\{\ga \in X: \ga \srp \xb\}$.
  \ep
\end{DEFINITION}

Because the order topology is coarser than any other topology with respect to which $\p$ is continuous, we have:

\begin{LEMMA}\label{can.do.order.top.1} $\p$ on $X$ is continuous with respect to a connected topology if and only if the order topology is connected.  $\p$ on $X$ is continuous with respect to a separable topology if and only if the order topology is separable.  \ep
\end{LEMMA}

\noi {\sc Proof of Lemma \ref{cont=CE}}.  Assume continuity of $\p$.  Then continuity of $\p$ on $X$ holds trivially.  For the CE condition, consider any $x = (x_1, \ldots, x_n)$.  We find a CE of $x$.  Assume, without loss of generality, that $x_1 \p \cdots \p x_n$.  By replication equivalence and monotonicity (Lemma \ref{monotonicity}), $x_1 \sim (x_1, \ldots, x_1) \p x \p (x_n, \ldots, x_n) \sim x_n$.  Hence the closed sets $\{\ga \elt X: \ga \p x\}$ and $\{\ga \elt X: \ga \rp x\}$ are nonempty.  Their union is $X$ and, by connectedness, they must intersect (in fact, at their infimum/supremum).  This intersection is a CE of $x$.  The CE condition holds.

Next assume the CE condition, and continuity of $\p$ on $X$.  Take an arbitrary $x$, with its certainty equivalent denoted CE.  Then $\{\ga \elt X:
\ga \succ x\} = \{\ga \elt X: \ga \succ CE\}$ and $\{\ga \elt X: \ga \srp x\}
= \{\ga \elt X: \ga \srp CE\}$ are open because $\p$ on $X$ is continuous.  Hence $\p$ (on $\SX$) is continuous.  \ep\vspace{\baselineskip}

Our continuity condition (taken from Gravel, Marchant, \& Sen 2008), is the closed sections condition of Fuhrken \& Richter (1991), but restricted to one dimension, which underscores its generality.  The above results show that it is essentially stronger than continuity of $\p$ on $X$, because the latter does not imply the CE condition, such as with lexicographic preferences with respect to
rank-ordered 
$n$-tuples from $\R^n$.  The following result is a corollary of Lemmas \ref{cont=CE} and \ref{can.do.order.top.1}.

\begin{LEMMA}\label{can.do.order.top.2} Assume the intuitive conditions.  $\p$ (on $\SX$) is continuous with respect to a connected topology on $X$ if and only if it is continuous with respect to the order topology of $\p$ on $X$ and the latter is connected.
  \ep
\end{LEMMA}

\noi Lemmas \ref{can.do.order.top.1} and \ref{can.do.order.top.2} show, effectively, that the topology taken on $X$ is a refinement of the order topology, and that it is immaterial which refinement it is.

It has sometimes been thought, erroneously, that a function representing a binary relation is always continuous with respect to the order topology.  However, any strictly increasing discontinuous function from $\R$ to $\R$ provides a counterexample.  The following 
well-known result (a corollary of Beardon \& Mehta 1994, Proposition 1; see also Steen \& Seebach Jr.\ 1970, pp.\ 67-68) gives details.

\begin{LEMMA}\label{U.noncont.gap} Assume that $\p$ on $X$ is continuous with respect to a connected topology, and that $U$ represents $\p$ on $X$.  Then the following statements are equivalent:
  \begin{enumerate}
  \item $U$ is continuous;
  \item $U(X)$ is an interval;
  \item $U(X)$ neither has a gap (maximal noncontained interval) of the form $(\gs, \gt]$ nor of the form $[\gs, \gt)$;
  \item $U(X)$ is a dense subset of its convex hull;
  \item For each pair of outcomes $\ga, \xg$, there exists a utility midpoint $\xb$ in the sense that $U(\xb) = (U(\ga) + U(\xg))/2$.  \ep
  \end{enumerate}
\end{LEMMA}\vspace{\baselineskip}

The following lemma provides the main step in the derivation of Theorem \ref{main.result.cont} from Theorem \ref{main.result}.

\begin{LEMMA}\label{cont.=>Arch} The intuitive conditions and continuity imply the Archimedean axiom.  \end{LEMMA}

\noi {\sc Proof}.  For contradiction, assume Archimedeanity violated.  That is, assume $x \succ y$ and $(nx,v) \srp (ny,w)$ for all $n$, with $\#x = \#y = k$ and $\#v = \#w = m$.  Let $\gd = CE(v), \xg = CE(w)$ (this counteralphabetic notation will be moist convenient).  $v\sim \gd \sim m\gd$ and $w \sim \xg \sim m\xg$ and strong associativity imply $(nx, v) \sim (nx, m\gd)$ and $(ny, w) \sim (ny, m\xg)$.  Hence we have $(nx,m\gd) \srp (ny,m\xg)$ for all $n$, which implies $(nmx,m\gd) \srp (nmy,m\xg)$ for all $n$, and then
%%%% 
\begin{equation}\label{nxdnyg} (nx,\gd) \srp (ny,\xg) \text{~for~all~} n.
\end{equation}
%%%%%%% 
Let $x \sim \ga, y \sim \xb$.  Then $x \sim k\ga$, $y \sim k\xb$, $nx \sim nk\ga$, and $ny \sim nk\xb$ imply, by strong associativity, $(nx,\gd) \sim (nk \ga,\gd)$ and $(ny,\xg) \sim (nk \xb, \xg)$ for all $n$.  Substituting this in Eq.\ \ref{nxdnyg} gives $(nk\ga,\gd) \srp (nk\xb,\xg)$ for all $n$.  By Lemma \ref{n.suff.large},
%%%%%%%%%%%% 
\begin{equation}\label{arch.outcomes} (n\ga,\gd) \srp (n\xb,\xg) \text{~for~all~} n.
\end{equation}
%%%%%%%%%%%% 
We have $\ga \succ \xb$.  By monotonicity, $\xg \succ \gd$.  Eq.\ \ref{arch.outcomes} suggests that the utility difference between $\ga$ and $\xb$ is infinitesimal relative to that between $\xg$ and $\gd$.

We first define a set $C$, an arc between $\xg$ and $\gd$ isomorphic to ${[}0,1]$.  For each rational number $q$ between 0 and 1, we choose some $m$ and $n$ with $q = m/n$.  We then define $q\xg + (1-q)\gd$, of $q\xg$ for short, as a CE of $(m\xg, (n-m)\gd)$.  By replication equivalence, it does not depend on the particular $m$ and $n$ chosen.  Thus we have an ordered set isomorphic to the rational numbers between $0$ and $1$.  For each irrational number $0 < r <1$, the two sets $\cup_{q: q < r} \{\gr \in X: \gr \srp q\xg\}$ and $\cup_{q: q > r} \{\gr \in X: \gr \succ q\xg\}$ are open and nonempty.  By connectedness, there exists at least one outcome contained in neither set.  It can be seen that there usually are many, and that they are not all equivalent.  At any rate, we take one and define it as $r \xg$.  Define the function that assigns $r\xg$ to each $0 \leq r \leq 1$.  (Formally, this can be done using the choice axiom from mathematical logic.)  This ordered set is isomorphic to $[0,1]$ (it is a continuum), denoted $C$.

For each $\gs\in C$ we define a certainty equivalent $\gs_{\ga}$ of $(\ga, \gs)$ and $\gs_{\xb}$ of $(\xb, \gs)$.  It generates two sets $\{\gs_{\ga}: \gs \in C\}$ and $\{\gs_{\xb}: \gs \in C\}$.

\begin{LEMMA}\label{nonsep} For each $\gt \succ \gs \in C$ we have $\gt_{\xb} \succ \gs_{\ga}$.
\end{LEMMA}

\noi {\sc Proof}.  This proof will be ended by {\it QED}.  Because the rational numbers are dense in $[0,1]$, there are rational numbers $m/n$ and $(m-k)/ n$ (with $k>0$) such that $\gt \succ (m/n)\xg$ and $((m-k)/n)\xg \succ \gs$.  Then, by monotonicity, also $\gt_{\xb} \succ ((m/n)\xg)_{\xb}$ and $(((m-k)/n)\xg)_{\ga} \succ \gs_{\ga}$.  It suffices to prove $((m/n)\xg)_{\xb} \succ (((m-k)/n)\xg)_{\ga}$, or $((m/n)\xg, \xb) \succ (((m-k)/n)\xg, \ga)$, or
%%%%%%%%% 
\begin{equation}\label{do} (n((m/n)\xg), n\xb) \succ (n(((m-k)/n)\xg), n\ga).
\end{equation}
%%%%%%%%% 
Because $ (m\xg, (n-m)\gd) \sim (m/n)\xg \sim n((m/n)\xg)$, we have, by strong associativity, $(n((m/n)\xg), n\xb) \sim (m\xg, (n-m)\gd, n\xb)$.  Similarly, $(n(((m-k)/n)\xg), n\ga) \sim ((m-k)\xg, (n-m+k)\gd, n\ga)$.  We substitute these indifferences in Eq.\ \ref{do} and get, as sufficient to prove, $(m\xg, (n-m)\gd, n\xb) \succ ((m-k)\xg, (n-m+k)\gd, n\ga)$.  By expansion independence, dropping $n-k$ common coordinates, this is equivalent to $$(k\xg, n\xb) \succ (k\gd, n\ga).$$ We finally prove this.  By Eq.\ \ref{arch.outcomes}, $(\xg, n\xb) \succ (\gd, n\ga)$.  For induction, $(j\xg, n\xb) \succ (j\gd, n\ga)$ implies, by expansion independence and then monotonicity, $(\xg, j\xg, n\xb) \succ (\xg, j\gd, n\ga) \succ (\gd, j\gd, n\ga)$, or $((j+1)\xg, n\xb) \succ ((j+1)\gd, n\ga)$.  It follows by induction that $(k\xg, n\xb) \succ (k\gd, n\ga)$.  
{\it QED}\vspace{\baselineskip}

Every preference interval $\{\gr \in X: \gs_{\xb} \srp \gr \srp \gs_{\ga}\}$ is nonempty because it contains $CE(\gs_{\xb}, \gs_{\ga})$ ($\gs_{\xb} \sim (\gs_{\xb},\gs_{\xb}) \srp (\gs_{\xb}, \gs_{\ga}) \srp (\gs_{\ga},\gs_{\ga})$, mostly by monotonicity).  These preference intervals are all disjoint by Lemma \ref{nonsep}.  Hence these are uncountably many disjoint nonempty open sets, which cannot be because of topological separability.  Contradiction has resulted and Lemma \ref{cont.=>Arch} has been proved.

The beginning of this proof has in fact shown that under the CE condition the Archimedean axiom can be restricted to the case where $x,y,v,w$ are outcomes.
\ep\vspace{\baselineskip}
%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

\begin{LEMMA}\label{U.cont} Assume that average utility holds with utility $U$, and that continuity holds with respect to a connected topology on $X$.  Then $U$ is continuous.  \ep
\end{LEMMA}

\noi {\sc Proof}.  For all values $U(\ga)$ and $U(\xg)$ in the image of $U$, the midpoint $(U(\ga) + U\xg))/2$ is also contained in the image of $U$, because it is the utility of the CE of $(\ga, \xg)$, which exists by Lemma \ref{cont=CE}.  By Lemma \ref{U.noncont.gap}, $U$ is continuous.  \ep\vspace{\baselineskip}

\noi {\sc Proof of Theorem \ref{main.result.cont}}.  To derive (i)$\Ra$(ii), we assume Statement (i).  The first four conditions in (ii) follow from substitution.  Condition 5 follows by taking the topology on $X$ generated by $U$, i.e., by $\p$ on $X$.  $U(X)$ being an interval implies connectedness and separability of this topology.

We finally assume Statement (ii), and derive Statement (i), continuity of $U$, and the uniqueness result.  By Lemma \ref{cont.=>Arch}, Archimedeanity is satisfied.  By Theorem \ref{main.result}, we obtain an
average-utility representation.  By Lemma \ref{U.cont}, $U$ is continuous.  By Lemma \ref{U.noncont.gap}, $U(X)$ is an interval.  The uniqueness follows from Theorem \ref{main.result}.  \ep\vspace{\baselineskip}

If in Theorem \ref{main.result.cont} we replace continuity by simple continuity, then the nontrivial implication, (ii)$\Ra$(i), can easily be derived from Debreu (1960), along the line of Blackorby, Davidson, \& Donaldson (1977).  First, on each $X^n$ with $n \geq 3$ we then get an additive representation $\sum_{j=1}^{n}V_{j,n}(x_j)$, mainly because joint independence is what is often called (preferential) separability.  The function $V_{j,n}$ at this stage can depend on $n$.  For each $n$, the functions $V_{1,n}, \ldots, V_{n,n}$'s are identical, and we obtain a representation $\sum_{j=1}^{n}U_n(x_j)$ (Blackorby, Davidson, \& Donaldson 1977 Lemma 1; Blackorby, Bossert, \& Donaldson 2005 Theorem 4.7).  Replication equivalence (relating dimensions $k$ and $m$ to each other through dimension $km$) implies that $U_n$ is independent of $n$ (as in Blackorby, Bossert, \& Donaldson 2005, Theorems 4.19, 6.1, and 6.2), and (dropping the $n$) that average utility $\sum_{j=1}^{n}U(x_j)/n$ represents preference on the whole domain $\SX$.  A special advantage of our more general analysis is that we obtain a uniform generalization of results on generalized means (Corollary \ref{gen.M=>cont}).\vspace{2\baselineskip}

\noi {\Large \bf Appendix C: Further proofs}\vspace{\baselineskip}

\noi {\sc Proof of Lemma \ref{ind=>joint.ind}}.  Both for $r_1 = c_1$ and $r_1 = d_1$, $(r_1,x_2,\ldots,x_n) = (1/n) r_1 + ((n-1)/n) (x_2,\ldots,x_n)$ and $(r_1,y_2,\ldots,y_n) = (1/n) r_1 + ((n-1)/n) (y_2,\ldots,y_n)$.  By independence, the preference between $(r_1,x_2,\ldots,x_n)$ and $(r_1,y_2,\ldots,y_n)$ must agree with that between $(x_2,\ldots,x_n)$ and $(y_2,\ldots,y_n)$ for both $r_1$, implying joint independence.
\ep\vspace{\baselineskip}

\noi {\sc Proof of Observation \ref{gen.mean.vs.pref}}.  This follows from substitutions and monotonicity.  \ep\vspace{\baselineskip}

\noi {\sc Proof of Theorem \ref{generd.means}}.  The implication (i)$\Ra$(ii) follows from substitution.  We, hence, assume (ii) and derive (i) and continuity and uniqueness.  $\p$ satisfies the first four conditions in Statement (ii) in Theorem \ref{main.result.cont}.  Continuity of $\p$ on $X$ follows because it is the natural order on $\R$.  The topology, the Euclidean one, is connected and separable.  The CE condition follows from reflexivity.  By Theorem \ref{main.result.cont}, there exists a continous AU representation.  By monotonicity, $U$ is strictly increasing.  By idemtotence, $M$ is the CE function.  (i) has been proved.  The uniqueness follows from Theorem \ref{main.result.cont}.  \ep\vspace{\baselineskip}

\noi {\sc Proof of Lemma \ref{ass=>}}.  For idempotence, assume $M(n\ga) = \xb$.  By associativity, with $k=n$, we have $M(n\ga) = M(n\xb)$.  By monotonicity, $\ga = \xb$.

For replication equivalence (also derived by Blackorby, Bossert, \& Donaldson 2005 Theorem 4.20), take the certainty equivalent $M(x) = \ga$.  By $k$-fold application of associativity, $M(kx) = M(k(\#x\ga))$.  By idempotence, $M((k\#x)\ga) = \ga$.  We conclude that $M(x) = M(kx)$.

For joint independence, assume $(x_1, \ldots, x_n) \p (y_1, \ldots, y_n)$.  The CE condition is satisfied with $M$ the CE function, by idempotence.  Take certainty equivalents $M(x) = \xb$ and $M(y) = \xg$, respectively.  Then $\xb \geq \xg$.  By monotonicity, $(\ga, \xb, \ldots, \xb) \p (\ga, \xg, \ldots, \xg)$ (both
$n+1$-tuples).  By associativity, the former
$n+1$-tuple is indifferent to $(\ga, x_1, \ldots, x_n)$ and the latter to $(\ga, y_1, \ldots, y_n)$.  Hence $(\ga, x_1, \ldots, x_n) \p (\ga, y_1, \ldots, y_n)$.  Expansion independence has been shown.  By Lemma \ref{exp.sep.then.sep}, this implies joint independence of $\p$ and, hence, of $M$.  \ep\vspace{\baselineskip}

\noi {\sc Proof of Lemma \ref{av.=>repl.eq}}.  For induction, assume $kx \sim x$.  It implies $kx \p x$ and $kx \rp x$.  By averaging, $kx \p (k+1)x \p x$ and $kx \rp (k+1)x \rp x$.  $(k+1)x \sim x$ follows.  Induction implies replication equivalence.  \ep\vspace{\baselineskip}

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of These.''  {\it In\/} Wolfgang Eichhorn (ed.) {\it Measurement in
  Economics\/} (Theory and Applications of Economic Indices), 311--326,
Physica-Verlag, Heidelberg.

\hangafter= 1\hangindent=\parindent\noindent Wakker, Peter P. (1989) ``{\it
  Additive Representations of Preferences, A New Foundation of Decision
  Analysis\/}.'' Kluwer Academic Publishers, Dordrecht.

\end{document}



@@@@@@@@@@@@@@@@@@@@@@@@

rrr Kan geen natuurlijke plek vinden nog. Peleg, Bezalel \& Hans J.M. Peters
(2009) used AU for
n-tuples (1/n lotteries) over finite sets. \hangafter=
1\hangindent=\parindent\noindent Peleg, Bezalel \& Hans J.M. Peters (2009)
``Nash Consistent Representation of Effectivity Functions through Lottery
Models,'' {\it Games and Economic Behavior\/} 65, 503--515.





@@@

Lemma: Bisymmetry implies joint independence through Gorman's theorem. Hence
strong bisymmetry in Grabisch et al. (2011a p. 7) and of Marichal (2000@@
??rrr) can be weakened to the case of only $p=2$.


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